# Critical Phenomena and Transitions in Swollen Polymer Networks and in Linear Macromolecules

## Abstract

The theory of equilibrium swelling and change of dimensions in networks and linear macromolecules is reexamined with the primary object of elucidating the requirements for discontinuous shrinkage through a transition of first order in a poor solvent. The parameter x that characterizes the solvent-polymer interactions is represented as a function of composition by x = ×i + ×2V2 + ×3v22+ where v2is the volume fraction of polymer. For networks, the elastic free energy is formulated according to current theory of elasticity, which, in contrast to earlier theories, takes account of the nonaffine displacements of network junctions under strain. Coexistence of two phases differing in degree of swelling requires substantial contributions to x from higher terms in xj with J ≥ 2. The locus of critical points in the ×1,×2plane, for solvent activity a1= 1, is examined with xj = 0 for j > 2 and also for arbitrary values of X3. Required values of X2 are ≥ ⅓, the lower limit being reached for vanishing degrees of cross-linking, i.e., for 1/xc= 0, where xcis the (mean) number of segments in a network chain. As xcdecreases, the required value of ×2 increases and that of xi decreases commencing at X1 = ½ for l/xc = 0. Requirements for coexistence of two swollen phases in equilibrium with pure solvent (i.e., for a1= 1) also are explored. Nonpolar solvent-polymer mixtures for which the x parameters meet the indicated requirements are deemed to be rare. A small proportion of ionized groups on the polymer chain may induce discontinuous shrinkage of the swollen network via a phase transition, as Tanaka and others have shown. The collapse of a randomly coiled linear macromolecule in a poor solvent is susceptible to similar theoretical treatment. Values of the parameters xi and X2 required for critical conditions and associated transitions may fall within the range of those found experimentally in real polymer-solvent systems. In particular, the system cyclohexane-polystyrene appears to meet the requirements in this regard. Calculations show that the collapse should occur in this system over a range of ca. 2 °C for a molecular weight of ca. 107, in agreement with experiment. Aqueous DNA is an example having the attributes for such behavior as established definitively by the investigations of Post and Zimm. © 1986, American Chemical Society. All rights reserved.